In context, you can use the single word ‘conjugate’
instead of the two-word ‘complex conjugate’.
For example:
the conjugate of $\,3+2i\,$ is $\,3 - 2i\,$;
the conjugate of $\,3 - 2i\,$ is $\,3 + 2i\,$.

COMPLEX CONJUGATE PAIRS:
A complex number with a nonzero imaginary part, together with its conjugate,
are called a complex conjugate pair or, more simply, a conjugate pair.
Thus, $\,a + bi\,$ and $\,a - bi\,$ are a complex conjugate pair
for $\,b\ne 0\,$.

The numbers (say) $\,5 + 0i\,$ and $\,5 - 0i\,$ are complex conjugates,
but they are not a complex conjugate pair,
since they are not distinct numbers.

THE CONJUGATE OF A REAL NUMBER:
If $\,x\,$ is a real number, then $\,\overline{x} = x\,$.
That is, the complex conjugate of a real number is itself.
Proof:
$\overline{x} = \overline{x + 0i} = x - 0i = x$

THE CONJUGATE OF POWERS:
For integers $\,n\ge 2\,$ and for all complex numbers $\,z\ $:
$$
\overline{z^n} = {(\,\,\overline{z}\,)\,}^n
$$
Proof when $\,n = 2\,$:
Since the conjugate of a product is the product of the conjugates,
$$
\overline{z^2} = \overline{z\cdot z} = \overline{z}\cdot\overline{z} = {(\,\,\overline{z}\,)\,}^2
$$
The extension to higher powers is discussed next.

A complex number and its conjugate:

$z := a + bi$

$\overline{z} = a - bi$

A complex number $\,z\,$ can be represented:

$\bullet$ as a point

$\bullet$ as a vector from the origin to the point

The length of the vector representing $\,z\,$
is called the modulus of $\,z\,$
and is denoted by $\,|z|\,$.

Equivalently, $\,|z|\,$ represents
the distance of the complex number
from the origin in the complex plane.

From the Pythagorean Theorem:
$|z| = \sqrt{a^2 + b^2}$

Extending Properties to Finite Sums/Products

Properties that ‘look like’ they hold for only two complex numbers actually hold
for any finite number of complex numbers.
The key is to take two things and treat them as a single thing, as shown next:
$$
\begin{alignat}{2}
\overline{u + v + w} \ &= \overline{(u + v) + w} &&\qquad\text{treat $\,u+v\,$ as a singleton}\cr\cr
&= \overline{u+v} + \overline{w} &&\qquad\text{it works for two}\cr\cr
&= \overline{u} + \overline{v} + \overline{w} &&\qquad\text{it works for two}
\end{alignat}
$$
Since it works for two, it works for three.
Repeating the processsince it works for three, it works for four.
And so on!

Thus:

the conjugate of any finite sum of complex numbers is the sum of the conjugates

the conjugate of any finite product of complex numbers is the product of the conjugates

since $\,z^n\,$ is a finite product for any positive integer $\,n\,$, this ‘treat it as a singleton’
technique shows that $\,\overline{z^n} = {(\,\,\overline{z}\,)}^n\,$

An Important Property of Polynomials with Real Coefficients

A CONJUGATE PROPERTYfor polynomials with real coefficients

If $\,P\,$ is a polynomial with real number coefficients, then $\,\overline{P(z)} = P(\overline z)\,$.

IDEA OF PROOF:
The ideas that make this work are illustrated with a quadratic polynomial, but
it should be clear that they work for a polynomial of any degree.